Logarithms: the idea

Converting between index form and logarithm form.

Indices & Logarithms · O/L Maths

Level 1: The idea of a logarithm

Link powers of 10 with logarithms

A logarithm answers the question "which power?". Since 10³ = 1000, the logarithm of 1000 to base 10 is 3.

10³ = 1000 ⇔ log₁₀ 1000 = 3

bⁿ = x ⇔ log_b x = n

• log₁₀ 10000
• Which power of 10 is 10 000? 10⁴ = 10 000
• log₁₀ 10000 = 4

Level 2: Other bases and halving

Use log₂, log₃ … and switch between forms

Logarithms can use any base, not only 10: log₂ 8 = 3 (2³ = 8), log₃ 81 = 4 (3⁴ = 81).

Move easily between logarithm form and index form.

• log₂ x = 5
• Write in index form: x = 2⁵
• x = 32

Level 3: Special values and negative powers of 10

log 1 = 0, log b = 1 and negative logs

log_b 1 = 0 log_b b = 1

Numbers smaller than 1 have negative logarithms: log₁₀ 0.01 = log₁₀ 10⁻² = −2.

• log₁₀ 0.001
• 0.001 = 1 ÷ 1000 = 10⁻³
• log₁₀ 0.001 = −3

Level 4: Challenge

Mixed logarithm problems

In every problem, ask "which power?". Negative case: log₂ (1)/(8) = −3.

• log₂ 8 + log₃ 9
• log₂ 8 = 3 (2³ = 8)
• log₃ 9 = 2 (3² = 9)
• 3 + 2 = 5